where \(\left\lbrace  \xi _{l}\right\rbrace  \)  is an infinite sequence decreasing
in \(l\) . The block sampling increases the efficiency with respect
to the original algorithm since \(u\)  and \(w\)  are strongly correlated,
while the introduction of \(\xi _{l}\)  reduces the sampling of useless
\(w_{j}s\) .
In what follows, we will use a deterministic, decreasing sequence
\(\left\lbrace  \xi _{j}\right\rbrace  _{j\in \mathbb {N}}\)  but, in general, a random
sequence could also be considered. [1]}
found that the mixing of the resulting Markov chain depends on the
rate of increase of \(\frac{E\left[w_{j}\right]}{\xi _{j}}\) : higher
rates of increase are associated with better mixing but longer running
times, since the average size of the sets \(\left\lbrace  j\left|w_{j}>u\right.\right\rbrace  \) 
increases. They suggest increasing the rate of increase of \(\frac{E\left[w_{j}\right]}{\xi _{j}}\) 
until the gains in mixing are counter-balanced by the longer running
time. In their examples, [1]} find
that \(\frac{E\left[w_{j}\right]}{\xi _{j}}\propto \left(\frac{3}{2}\right)^{j}\) 
strikes a good balance. Thus we set \(\xi _{j}\propto \frac{E\left[w_{j}\right]}{\left(1.5\right)^{j}}\) . We have \(\mathbf {x}_{t}\oslash \mathbf {\mu }_{t}=\mathbf {\varepsilon }_{t}\)  and \(f_{\mathbf {\varepsilon }_{t}}\left(\cdot \right)=\underset{{\scriptstyle j=1}}{\overset{{\scriptstyle \infty }}{\sum }}w_{j}\text{logN}_{d}\left(\cdot \left|\mathbf {m}_{j},\mathbf {\Sigma }{}_{j}\right.\right)\) ,
then

\(\begin{aligned}f_{\mathbf {x}_{t}|\ldots }\left(\mathbf {x}\right) & =f_{\mathbf {\varepsilon }_{t}}\left(\mathbf {x}\oslash \mathbf {\mu }\right)\left|\frac{\partial \mathbf {\varepsilon }}{\partial \mathbf {x}}\right|=\\& =\underset{{\scriptstyle j=1}}{\overset{{\scriptstyle \infty }}{\sum }}w_{j}\text{logN}_{d}\left(\mathbf {x}\oslash \mathbf {\mu }\left|\mathbf {m}_{j},\mathbf {\Sigma }_{j}\right.\right)\overset{{\scriptstyle d}}{\underset{{\scriptstyle i=1}}{\prod }}\frac{1}{\mu _{i}}=\\& =\underset{{\scriptstyle j=1}}{\overset{{\scriptstyle \infty }}{\sum }}w_{j}\overset{{\scriptstyle d}}{\underset{{\scriptstyle i=1}}{\prod }}\frac{1}{\mu _{i}}\overset{{\scriptstyle d}}{\underset{{\scriptstyle i=1}}{\prod }}\frac{\mu _{i}}{x_{i}}\left(2\pi \right)^{-\frac{d}{2}}\left|\mathbf {\Sigma }_{j}\right|{}^{-\frac{1}{2}}\exp \left\lbrace  -\frac{1}{2}\left(\log \left(\mathbf {x}\oslash \mathbf {\mu }\right)-\mathbf {m}_{j}\right)^{^{\prime }}\mathbf {\Sigma }^{-1}\left(\log \left(\mathbf {x}\oslash \mathbf {\mu }\right)-\mathbf {m}_{j}\right)\right\rbrace  =\\& =\underset{{\scriptstyle j=1}}{\overset{{\scriptstyle \infty }}{\sum }}w_{j}\overset{{\scriptstyle d}}{\underset{{\scriptstyle i=1}}{\prod }}\frac{1}{x_{i}}\left(2\pi \right)^{-\frac{d}{2}}\left|\mathbf {\Sigma }_{j}\right|{}^{-\frac{1}{2}}\exp \left\lbrace  -\frac{1}{2}\left(\log \mathbf {x}-\log \mathbf {\mu }-\mathbf {m}_{j}\right)^{^{\prime }}\mathbf {\Sigma }_{j}^{-1}\left(\log \mathbf {x}-\log \mathbf {\mu }-\mathbf {m}_{j}\right)\right\rbrace  =\\& =\underset{{\scriptstyle j=1}}{\overset{{\scriptstyle \infty }}{\sum }}w_{j}\text{logN}_{d}\left(\mathbf {x}\left|\mathbf {m}_{j}+\log \mathbf {\mu },\mathbf {\Sigma }_{j}\right.\right)\end{aligned}\) 
